91Ó°ÊÓ

Prove that, if a particle moving with linear simple harmonic motion of amplitude \(a\) has velocity \(v\) when distant \(x\) from the centre of its path, then \(v=\omega \sqrt{a^{2}-x^{2}}\) where \(\omega\) is a constant. A point travelling with linear S.H.M. has speeds \(3 \mathrm{~ms}^{-1}\) and \(2 \mathrm{~ms}^{-1}\) when distant \(1 \mathrm{~m}\) and \(2 \mathrm{~m}\) respectively from the centre of oscillations. Calculate the amplitude, the periodic time and the maximum velocity.

A particle is attached to one end of a light elastic string, the other end of which is fastened to a fixed point \(\mathrm{A}\) on a smooth plane inclined at an angle arcsin \(\frac{1}{4}\) to the horizontal. The particle rests in equilibrium at a point \(O\) on the plane with the string stretched along a line of greatest slope and extended by an amount \(c .\) If the particle is released from rest at a point P on AO produced, show that so long as the string remains taut the particle will oscillate in simple harmonic motion about \(\mathrm{O}\) as centre, and state the periodic time. If \(\mathrm{OP}=2 c\), find the velocity of the particle when it first reaches \(\mathrm{O}\) after leaving \(\mathrm{P} .\)